Geometry and Dynamics in Gromov Hyperbolic Metric Spaces with an Emphasis on Non-Proper Settings
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Quasi-Hyperbolic Planes in Relatively Hyperbolic Groups
QUASI-HYPERBOLIC PLANES IN RELATIVELY HYPERBOLIC GROUPS JOHN M. MACKAY AND ALESSANDRO SISTO Abstract. We show that any group that is hyperbolic relative to virtually nilpotent subgroups, and does not admit peripheral splittings, contains a quasi-isometrically embedded copy of the hy- perbolic plane. In natural situations, the specific embeddings we find remain quasi-isometric embeddings when composed with the inclusion map from the Cayley graph to the coned-off graph, as well as when composed with the quotient map to \almost every" peripheral (Dehn) filling. We apply our theorem to study the same question for funda- mental groups of 3-manifolds. The key idea is to study quantitative geometric properties of the boundaries of relatively hyperbolic groups, such as linear connect- edness. In particular, we prove a new existence result for quasi-arcs that avoid obstacles. Contents 1. Introduction 2 1.1. Outline 5 1.2. Notation 5 1.3. Acknowledgements 5 2. Relatively hyperbolic groups and transversality 6 2.1. Basic definitions 6 2.2. Transversality and coned-off graphs 8 2.3. Stability under peripheral fillings 10 3. Separation of parabolic points and horoballs 12 3.1. Separation estimates 12 3.2. Embedded planes 15 3.3. The Bowditch space is visual 16 4. Boundaries of relatively hyperbolic groups 17 4.1. Doubling 19 4.2. Partial self-similarity 19 5. Linear Connectedness 21 Date: April 23, 2019. 2000 Mathematics Subject Classification. 20F65, 20F67, 51F99. Key words and phrases. Relatively hyperbolic group, quasi-isometric embedding, hyperbolic plane, quasi-arcs. The research of the first author was supported in part by EPSRC grants EP/K032208/1 and EP/P010245/1. -
EMBEDDINGS of GROMOV HYPERBOLIC SPACES 1 Introduction
GAFA, Geom. funct. anal. © Birkhiiuser Verlag, Basel 2000 Vol. 10 (2000) 266 - 306 1016-443X/00/020266-41 $ 1.50+0.20/0 I GAFA Geometric And Functional Analysis EMBEDDINGS OF GROMOV HYPERBOLIC SPACES M. BONK AND O. SCHRAMM Abstract It is shown that a Gromov hyperbolic geodesic metric space X with bounded growth at some scale is roughly quasi-isometric to a convex subset of hyperbolic space. If one is allowed to rescale the metric of X by some positive constant, then there is an embedding where distances are distorted by at most an additive constant. Another embedding theorem states that any 8-hyperbolic met ric space embeds isometrically into a complete geodesic 8-hyperbolic space. The relation of a Gromov hyperbolic space to its boundary is fur ther investigated. One of the applications is a characterization of the hyperbolic plane up to rough quasi-isometries. 1 Introduction The study of Gromov hyperbolic spaces has been largely motivated and dominated by questions about Gromov hyperbolic groups. This paper studies the geometry of Gromov hyperbolic spaces without reference to any group or group action. One of our main theorems is Embedding Theorem 1.1. Let X be a Gromov hyperbolic geodesic metric space with bounded growth at some scale. Then there exists an integer n such that X is roughly similar to a convex subset of hyperbolic n-space lHIn. The precise definitions appear in the body of the paper, but now we briefly discuss the meaning of the theorem. The condition of bounded growth at some scale is satisfied, for example, if X is a bounded valence graph, or a Riemannian manifold with bounded local geometry. -
Arxiv:1612.03497V2 [Math.GR] 18 Dec 2017 2-Sphere Is Virtually a Kleinian Group
BOUNDARIES OF DEHN FILLINGS DANIEL GROVES, JASON FOX MANNING, AND ALESSANDRO SISTO Abstract. We begin an investigation into the behavior of Bowditch and Gro- mov boundaries under the operation of Dehn filling. In particular we show many Dehn fillings of a toral relatively hyperbolic group with 2{sphere bound- ary are hyperbolic with 2{sphere boundary. As an application, we show that the Cannon conjecture implies a relatively hyperbolic version of the Cannon conjecture. Contents 1. Introduction 1 2. Preliminaries 5 3. Weak Gromov–Hausdorff convergence 12 4. Spiderwebs 15 5. Approximating the boundary of a Dehn filling 20 6. Proofs of approximation theorems for hyperbolic fillings 23 7. Approximating boundaries are spheres 39 8. Ruling out the Sierpinski carpet 42 9. Proof of Theorem 1.2 44 10. Proof of Corollary 1.4 45 Appendix A. δ{hyperbolic technicalities 46 References 50 1. Introduction One of the central problems in geometric group theory and low-dimensional topology is the Cannon Conjecture (see [Can91, Conjecture 11.34], [CS98, Con- jecture 5.1]), which states that a hyperbolic group whose (Gromov) boundary is a arXiv:1612.03497v2 [math.GR] 18 Dec 2017 2-sphere is virtually a Kleinian group. By a result of Bowditch [Bow98] hyperbolic groups can be characterized in terms of topological properties of their action on the boundary. The Cannon Conjecture is that (in case the boundary is S2) this topological action is in fact conjugate to an action by M¨obiustransformations. Rel- atively hyperbolic groups are a natural generalization of hyperbolic groups which are intended (among other things) to generalize the situation of the fundamental group of a finite-volume hyperbolic n-manifold acting on Hn. -
15 Nov 2008 3 Ust Nsmercsae N Atcs41 38 37 Lattices and Spaces Symmetric in References Subsets Graph Pants the 13
THICK METRIC SPACES, RELATIVE HYPERBOLICITY, AND QUASI-ISOMETRIC RIGIDITY JASON BEHRSTOCK, CORNELIA DRUT¸U, AND LEE MOSHER Abstract. We study the geometry of nonrelatively hyperbolic groups. Gen- eralizing a result of Schwartz, any quasi-isometric image of a non-relatively hyperbolic space in a relatively hyperbolic space is contained in a bounded neighborhood of a single peripheral subgroup. This implies that a group be- ing relatively hyperbolic with nonrelatively hyperbolic peripheral subgroups is a quasi-isometry invariant. As an application, Artin groups are relatively hyperbolic if and only if freely decomposable. We also introduce a new quasi-isometry invariant of metric spaces called metrically thick, which is sufficient for a metric space to be nonhyperbolic rel- ative to any nontrivial collection of subsets. Thick finitely generated groups include: mapping class groups of most surfaces; outer automorphism groups of most free groups; certain Artin groups; and others. Nonuniform lattices in higher rank semisimple Lie groups are thick and hence nonrelatively hy- perbolic, in contrast with rank one which provided the motivating examples of relatively hyperbolic groups. Mapping class groups are the first examples of nonrelatively hyperbolic groups having cut points in any asymptotic cone, resolving several questions of Drutu and Sapir about the structure of relatively hyperbolic groups. Outside of group theory, Teichm¨uller spaces for surfaces of sufficiently large complexity are thick with respect to the Weil-Peterson metric, in contrast with Brock–Farb’s hyperbolicity result in low complexity. Contents 1. Introduction 2 2. Preliminaries 7 3. Unconstricted and constricted metric spaces 11 4. Non-relative hyperbolicity and quasi-isometric rigidity 13 5. -
Inducing Maps Between Gromov Boundaries
INDUCING MAPS BETWEEN GROMOV BOUNDARIES J. DYDAK AND Z.ˇ VIRK Abstract. It is well known that quasi-isometric embeddings of Gromov hy- perbolic spaces induce topological embeddings of their Gromov boundaries. A more general question is to detect classes of functions between Gromov hyperbolic spaces that induce continuous maps between their Gromov bound- aries. In this paper we introduce the class of visual functions f that do induce continuous maps f˜ between Gromov boundaries. Its subclass, the class of ra- dial functions, induces H¨older maps between Gromov boundaries. Conversely, every H¨older map between Gromov boundaries of visual hyperbolic spaces induces a radial function. We study the relationship between large scale prop- erties of f and small scale properties of f˜, especially related to the dimension theory. In particular, we prove a form of the dimension raising theorem. We give a natural example of a radial dimension raising map and we also give a general class of radial functions that raise asymptotic dimension. arXiv:1506.08280v1 [math.MG] 27 Jun 2015 Date: May 11, 2018. 2010 Mathematics Subject Classification. Primary 53C23; Secondary 20F67, 20F65, 20F69. Key words and phrases. Gromov boundary, hyperbolic space, dimension raising map, visual metric, coarse geometry. This research was partially supported by the Slovenian Research Agency grants P1-0292-0101. 1 2 1. Introduction Gromov boundary (as defined by Gromov) is one of the central objects in the geometric group theory and plays a crucial role in the Cannon’s conjecture [7]. It is a compact metric space that represents an image at infinity of a hyperbolic metric space. -
On Haagerup and Kazhdan Properties Yves De Cornulier
On Haagerup and Kazhdan Properties Thèse de doctorat présentée à la Faculté des Sciences de Base Section de Mathématiques École Polytechnique Fédérale de Lausanne pour l’obtention du grade de Docteur ès Sciences par Yves de Cornulier de nationalité française titulaire d’un DEA de Mathématiques, Université Paris VII le 16 Décembre 2005 devant le jury composé de Peter Buser, directeur Alain Valette, codirecteur Éva Bayer Fluckiger, présidente du jury Laurent Bartholdi, rapporteur interne Étienne Ghys, rapporteur externe Pierre de la Harpe, rapporteur externe Lausanne, EPFL 2005 Remerciements Je voudrais avant tout remercier Alain Valette. Sa disponibilité et ses encoura- gements ont eu un rôle décisif dans ce travail. Je remercie chaleureusement Peter Buser, qui m’a permis de bénéficier des condi- tions de travail les plus enviables. Je remercie Eva Bayer d’avoir accepté de présider le jury. Je remercie égale- ment les rapporteurs Laurent Bartholdi, Étienne Ghys et Pierre de la Harpe aussi bien pour avoir accepté de participer au jury que pour de nombreuses discussions mathématiques durant cette thèse. Je remercie aussi tous les autres avec qui j’ai eu l’occasion de discuter de maths. Avant tout je remercie Romain Tessera, dont le rare esprit d’ouverture m’a permis de discuter de mille et un sujets (dont au moins neuf cent nonante-neuf mathématiques). Je remercie Bachir Bekka, qui m’a invité à deux reprises durant cette thèse (à Metz et à Rennes). Je remercie également Herbert Abels, Guillaume Aubrun, Yves Benoist, Emmanuel Breuillard, Gaëtan Chenevier, Yves Coudène, François Guéritaud, Olivier Guichard, Vincent Lafforgue, David Madore, Michel Matthey (pour qui j’ai une pensée émue), Andrés Navas, Yann Ollivier, Pierre Pansu, Frédéric Paulin, Bertrand Rémy, Joël Riou, Jean-François Quint, Gabriel Sabbagh, Yehuda Shalom, Olivier Wittenberg, et d’autres encore. -
Introduction to Hyperbolic Metric Spaces
Introduction to Hyperbolic Metric Spaces Adriana-Stefania Ciupeanu University of Manitoba Department of Mathematics November 3, 2017 A. Ciupeanu (UofM) Introduction to Hyperbolic Metric Spaces November 3, 2017 1 / 36 Introduction Introduction Geometric group theory is relatively new, and became a clearly identifiable branch of mathematics in the 1990s due to Mikhail Gromov. Hyperbolicity is a centre theme and continues to drive current research in the field. Geometric group theory is bases on the principle that if a group acts as symmetries of some geometric object, then we can use geometry to understand the group. A. Ciupeanu (UofM) Introduction to Hyperbolic Metric Spaces November 3, 2017 2 / 36 Introduction Introduction Gromov’s notion of hyperbolic spaces and hyperbolic groups have been studied extensively since that time. Many well-known groups, such as mapping class groups and fundamental groups of surfaces with cusps, do not meet Gromov’s criteria, but nonetheless display some hyperbolic behaviour. In recent years, there has been interest in capturing and using this hyperbolic behaviour wherever and however it occurs. A. Ciupeanu (UofM) Introduction to Hyperbolic Metric Spaces November 3, 2017 3 / 36 Introduction If Euclidean geometry describes objects in a flat world or a plane, and spherical geometry describes objects on the sphere, what world does hyperbolic geometry describe? Hyperbolic geometry takes place on a curved two dimensional surface called hyperbolic space. The essential properties of the hyperbolic plane are abstracted to obtain the notion of a hyperbolic metric space, which is due to Gromov. A. Ciupeanu (UofM) Introduction to Hyperbolic Metric Spaces November 3, 2017 4 / 36 Introduction Hyperbolic geometry is a non-Euclidean geometry, where the parallel postulate of Euclidean geometry is replaced with: For any given line R and point P not on R, in the plane containing both line R and point P there are at least two distinct lines through P that do not intersect R. -
A Crash Introduction to Gromov Hyperbolic Spaces
Language of metric spaces Gromov hyperbolicity Gromov boundary Conclusion A crash introduction to Gromov hyperbolic spaces Valentina Disarlo Universität Heidelberg Valentina Disarlo A crash introduction to Gromov hyperbolic spaces Language of metric spaces Gromov hyperbolicity Gromov boundary Conclusion Geodesic metric spaces Definition A metric space (X; d) is proper if for every r > 0 the ball B(x; r) is compact. It is geodesic if every two points of X are joined by a geodesic. n R with the Euclidean distance dEucl. the infinite tree T with its length distance (every edge has length 1); x y w Figure: The infinite tree T Valentina Disarlo A crash introduction to Gromov hyperbolic spaces Language of metric spaces Gromov hyperbolicity Gromov boundary Conclusion n Geodesic metric spaces: the hyperbolic space H n Disk model D n n 4 D ∶= {x ∈ R SSxS < 1} with the Riemannian metric induced by gx ∶= gEucl (1 − SSxSS2)2 n Upper half plane H n n 1 H ∶= {(x1;:::; xn) ∈ R S xn > 0} with the Riemannian metric induced by gx ∶= gEucl xn n ∀x; y ∈ D there exists a unique geodesic xy every geodesic segment can be extended indefinitely 2 2 Figure: radii/arcs ⊥ @D and half-circles/lines ⊥ @H Valentina Disarlo A crash introduction to Gromov hyperbolic spaces Language of metric spaces Gromov hyperbolicity Gromov boundary Conclusion n Geodesic metric spaces: the hyperbolic space H n The boundary of H is defined as the space: n n @H = { geodesic rays c ∶ [0; ∞) → H }~ ∼ ′ ′ c ∼ c if and only if d(c(t); c (t)) < M n n−1 n n It can be topologized so that @H = S and Hn = H ∪ @H is compact. -
Cannon-Thurston Maps for Trees of Hyperbolic Metric Spaces
CANNON-THURSTON MAPS FOR TREES OF HYPERBOLIC METRIC SPACES MAHAN MITRA Abstract. Let (X,d) be a tree (T) of hyperbolic metric spaces satisfying the quasi-isometrically embedded condition. Let v be a vertex of T . Let (Xv, dv) denote the hyperbolic metric space corresponding to v. Then i : Xv → X extends continuously to a map ˆi : Xv → X. This generalizes and gives a new proof of a Theorem of Cannon and Thurston. The techniques are used to give a differentc proofb of a result of Minsky: Thurston’s ending lamination conjecture for certain Kleinian groups. Applications to graphs of hyperbolic groups and local connectivity of limit sets of Kleinian groups are also given. 1. Introduction Let G be a hyperbolic group in the sense of Gromov [13]. Let H be a hyperbolic subgroup of G. We choose a finite symmetric generating set for H and extend it to a finite symmetric generating set for G. Let ΓH and ΓG denote the Cayley graphs of H, G respectively with respect to these generating sets. By adjoining the Gromov boundaries ∂ΓH and ∂ΓG to ΓH and ΓG, one obtains their compactifications ΓH and ΓG respectively. We’d like to understand the extrinsic geometry of H indG. Sinced the objects of study here come under the purview of coarse geometry, asymptotic or ‘large-scale’ information is of crucial importance. That is to say, one would like to know what happens ‘at infinity’. We put this in the more general context of a hyperbolic group H acting freely and properly discontinuously by isometries on a proper hyperbolic metric space X. -
Perspectives on Geometric Analysis
Surveys in Differential Geometry X Perspectives on geometric analysis Shing-Tung Yau This essay grew from a talk I gave on the occasion of the seventieth anniversary of the Chinese Mathematical Society. I dedicate the lecture to the memory of my teacher S.S. Chern who had passed away half a year before (December 2004). During my graduate studies, I was rather free in picking research topics. I[731] worked on fundamental groups of manifolds with non-positive curva- ture. But in the second year of my studies, I started to look into differential equations on manifolds. However, at that time, Chern was very much inter- ested in the work of Bott on holomorphic vector fields. Also he told me that I should work on Riemann hypothesis. (Weil had told him that it was time for the hypothesis to be settled.) While Chern did not express his opinions about my research on geometric analysis, he started to appreciate it a few years later. In fact, after Chern gave a course on Calabi’s works on affine geometry in 1972 at Berkeley, S.Y. Cheng told me about these inspiring lec- tures. By 1973, Cheng and I started to work on some problems mentioned in Chern’s lectures. We did not realize that the great geometers Pogorelov, Calabi and Nirenberg were also working on them. We were excited that we solved some of the conjectures of Calabi on improper affine spheres. But soon after we found out that Pogorelov [563] published his results right be- fore us by different arguments. Nevertheless our ideas are useful in handling other problems in affine geometry, and my knowledge about Monge-Amp`ere equations started to broaden in these years. -
The Ergodic Theory of Hyperbolic Groups
Contemporary Math,.ma.tlca Volume 597, 2013 l\ttp :/ /dX. dOl .org/10. 1090/CO!lll/597/11762 The Ergodic Theory of Hyperbolic Groups Danny Calegari ABSTRACT. Thll!le notes are a self-contained introduction to the use of dy namical and probabilistic methods in the study of hyperbolic groups. Moat of this material is standard; however some of the proofs given are new, and some results are proved in greater generality than have appeared in the literature. CONTENTS 1. Introduction 2. Hyperbolic groups 3. Combings 4. Random walks Acknowledgments 1. Introduction These are notes from a minicourse given at a workshop in Melbourne July 11- 15 2011. There is little pretension to originality; the main novelty is firstly that we a new (and much shorter) proof of Coornaert's theorem on Patterson-Sullivan measures for hyperbolic groups (Theorem 2.5.4), and secondly that we explain how to combine the results of Calegari-Fujiwara in [8] with that of Pollicott-Sharp [35] to prove central limit theorems for quite general classes functions on hyperbolic groups (Corollary 3.7.5 and Theorem 3.7.6), crucially without the hypothesis that the Markov graph encoding an automatic structure is A final section on random walks is much rnore cursory. 2. Hyperbolic groups 2.1. Coarse geometry. The fundamental idea in geometric group theory is to study groups as automorphisms of geometric spaces, and as a special case, to study the group itself (with its canonical self-action) as a geometric space. This is accomplished most directly by means of the Cayley graph construction. -
Curvature-Free Margulis Lemma for Gromov-Hyperbolic Spaces
Curvature-Free Margulis Lemma for Gromov-Hyperbolic Spaces G. Besson, G. Courtois, S. Gallot, A. Sambusetti December 2, 2020 Abstract We prove curvature-free versions of the celebrated Margulis Lemma. We are interested by both the algebraic aspects and the geometric ones, with however an emphasis on the second and we aim at giving quantitative (computable) estimates of some important invariants. Our goal is to get rid of the pointwise curvature assumptions in order to extend the results to more general spaces such as certain metric spaces. Essentially the upper bound on the curvature is replaced by the assumption that the space is hyperbolic in the sense of Gromov and the lower bound of the curvature by an upper bound on the entropy of which we recall the definition. Contents 1 Introduction 2 2 Basic definitions and notations8 3 Entropy, Doubling and Packing Properties 10 3.1 Entropies . 10 3.2 Doubling and Packing Properties . 13 3.3 Comparison between the various possible hypotheses : bound on the entropy, doubling and packing conditions, Ricci curvature . 14 3.4 Doubling property induced on subgroups . 20 4 Free Subgroups 21 4.1 Ping-pong Lemma . 21 arXiv:1712.08386v3 [math.DG] 1 Dec 2020 4.2 When the asymptotic displacement is bounded below . 24 4.3 When some Margulis constant is bounded below: . 27 4.3.1 When the Margulis constant L∗ is bounded below: . 28 4.3.2 When the Margulis constant L is bounded below: . 32 4.4 Free semi-groups for convex distances . 35 5 Co-compact actions on Gromov-hyperbolic spaces 36 5.1 A Bishop-Gromov inequality for Gromov-hyperbolic spaces .